Abstract
Causality places nontrivial constraints on QFT in Lorentzian signature, for
example fixing the signs of certain terms in the low energy Lagrangian. In
d-dimensional conformal field theory, we show how such constraints are encoded
in crossing symmetry of Euclidean correlators, and derive analogous constraints
directly from the conformal bootstrap (analytically). The bootstrap setup is a
Lorentzian four-point function corresponding to propagation through a
shockwave. Crossing symmetry fixes the signs of certain log terms that appear
in the conformal block expansion, which constrains the interactions of
low-lying operators. As an application, we use the bootstrap to rederive the
well known sign constraint on the \$(\partial\phi)^4\$ coupling in effective
field theory, from a dual CFT. We also find constraints on theories with higher
spin conserved currents. Our analysis is restricted to scalar correlators, but
we argue that similar methods should also impose nontrivial constraints on the
interactions of spinning operators.
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